Exercise 19, § 2

Let $k$ be a commutative field of characteristic $\neq 2$ and let Q be the quadratic form $x_1x_3 + x_2^2$ on $k^3$. Show that the group $\mathbf{SO}(Q)$ acts doubly transitively on the set of isotropic lines in $k^3$. Compare the corresponding Tits system (Exerc. 7) with that obtained in no. 2 for $n = 2$ (cf. Algebra Chap. IX, § 9, Exerc. 15).